Suggestion for notation of a sequence

Question

What could be a good notation for a sequence $(x_n)$ chosen from a vector space $\displaystyle V= \bigcup_{n\in \mathbb N} f_n(V_n)$ where all $V_n$ are vector spaces and $f_n$ are linear maps.

Answer 1

Using $n$ for both the index in $x_n$ and to index the union which defined $V$ doesn't seem like a good idea. Here's why: if $x_n \in\bigcup_{m \in \Bbb N} f_m\big(V_m\big)$, there is $m(n) \in \Bbb N$ such that $x_n \in f_{m(n)}\big(V_{m(n)}\big)$. This means that there is $v_{m(n)} \in V_{m(n)}$ such that $x_n = f_{m(n)}\big(v_{m(n)}\big)$. You want to be able to distinguish things. So $$(x_n)_{n \in \Bbb N} = \big(\,f_{m(n)}\big(v_{m(n)}\big)\big)_{n \in \Bbb N},$$and the function $m\colon \Bbb N \to \Bbb N$ depends on the sequence itself.